Syringe length/injection depth?

Hello, I initially started on compounded vials from my local pharmacy working my way up to the 15mg dose I’m on now. They gave me syringes with a fairly long needle on them. I injected almost all the way as deep as the needle would go.

Now I’ve switched to Lily Direct single dose vials and the needles they send are much shorter. Which got me thinking, does the depth of the injection matter? If I’m injecting into a fatty layer of tissue with the shorter needles would that change how the medicine enters and disperses through my body or however it gets to wherever it needs to be to work as intended?

Or if it needs to be injected into muscle below the fat wouldn’t we need a longer tip on the syringe?

Thanks for any insight!